Calculus Examples

$\int_{- 1}^{1} x^{2} + 2 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 1}^{1} x^{2} d x + \int_{- 1}^{1} 2 d x$

Step 2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{- 1}^{1} + \int_{- 1}^{1} 2 d x$

Step 3

Apply the constant rule.

$\frac{1}{3} x^{3}]_{- 1}^{1} + 2 x]_{- 1}^{1}$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Combine $\frac{1}{3} x^{3}]_{- 1}^{1}$ and $2 x]_{- 1}^{1}$ .

$\frac{1}{3} x^{3} + 2 x]_{- 1}^{1}$

Step 4.2

Substitute and simplify.

Tap for more steps...

Step 4.2.1

Evaluate $\frac{1}{3} x^{3} + 2 x$ at $1$ and at $- 1$ .

$(\frac{1}{3} \cdot 1^{3} + 2 \cdot 1) - (\frac{1}{3} {(- 1)}^{3} + 2 \cdot - 1)$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

One to any power is one.

$\frac{1}{3} \cdot 1 + 2 \cdot 1 - (\frac{1}{3} {(- 1)}^{3} + 2 \cdot - 1)$

Step 4.2.2.2

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3} + 2 \cdot 1 - (\frac{1}{3} {(- 1)}^{3} + 2 \cdot - 1)$

Step 4.2.2.3

Multiply $2$ by $1$ .

$\frac{1}{3} + 2 - (\frac{1}{3} {(- 1)}^{3} + 2 \cdot - 1)$

Step 4.2.2.4

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} + 2 \cdot \frac{3}{3} - (\frac{1}{3} {(- 1)}^{3} + 2 \cdot - 1)$

Step 4.2.2.5

Combine $2$ and $\frac{3}{3}$ .

$\frac{1}{3} + \frac{2 \cdot 3}{3} - (\frac{1}{3} {(- 1)}^{3} + 2 \cdot - 1)$

Step 4.2.2.6

Combine the numerators over the common denominator.

$\frac{1 + 2 \cdot 3}{3} - (\frac{1}{3} {(- 1)}^{3} + 2 \cdot - 1)$

Step 4.2.2.7

Simplify the numerator.

Tap for more steps...

Step 4.2.2.7.1

Multiply $2$ by $3$ .

$\frac{1 + 6}{3} - (\frac{1}{3} {(- 1)}^{3} + 2 \cdot - 1)$

Step 4.2.2.7.2

Add $1$ and $6$ .

$\frac{7}{3} - (\frac{1}{3} {(- 1)}^{3} + 2 \cdot - 1)$

Step 4.2.2.8

Raise $- 1$ to the power of $3$ .

$\frac{7}{3} - (\frac{1}{3} \cdot - 1 + 2 \cdot - 1)$

Step 4.2.2.9

Combine $\frac{1}{3}$ and $- 1$ .

$\frac{7}{3} - (\frac{- 1}{3} + 2 \cdot - 1)$

Step 4.2.2.10

Move the negative in front of the fraction.

$\frac{7}{3} - (- \frac{1}{3} + 2 \cdot - 1)$

Step 4.2.2.11

Multiply $2$ by $- 1$ .

$\frac{7}{3} - (- \frac{1}{3} - 2)$

Step 4.2.2.12

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7}{3} - (- \frac{1}{3} - 2 \cdot \frac{3}{3})$

Step 4.2.2.13

Combine $- 2$ and $\frac{3}{3}$ .

$\frac{7}{3} - (- \frac{1}{3} + \frac{- 2 \cdot 3}{3})$

Step 4.2.2.14

Combine the numerators over the common denominator.

$\frac{7}{3} - \frac{- 1 - 2 \cdot 3}{3}$

Step 4.2.2.15

Simplify the numerator.

Tap for more steps...

Step 4.2.2.15.1

Multiply $- 2$ by $3$ .

$\frac{7}{3} - \frac{- 1 - 6}{3}$

Step 4.2.2.15.2

Subtract $6$ from $- 1$ .

$\frac{7}{3} - \frac{- 7}{3}$

Step 4.2.2.16

Move the negative in front of the fraction.

$\frac{7}{3} - - \frac{7}{3}$

Step 4.2.2.17

Multiply $- 1$ by $- 1$ .

$\frac{7}{3} + 1 (\frac{7}{3})$

Step 4.2.2.18

Multiply $\frac{7}{3}$ by $1$ .

$\frac{7}{3} + \frac{7}{3}$

Step 4.2.2.19

Combine the numerators over the common denominator.

$\frac{7 + 7}{3}$

Step 4.2.2.20

Add $7$ and $7$ .

$\frac{14}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{14}{3}$

Decimal Form:

$4.\overline{6}$

Mixed Number Form:

$4 \frac{2}{3}$

Step 6